3.555 \(\int \frac{\sqrt{a+b x} \sqrt{c+d x}}{x^5} \, dx\)

Optimal. Leaf size=256 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right )}{96 a^2 c^2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{192 a^3 c^3 x}+\frac{\left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{24 a c x^3}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 x^4} \]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(4*x^4) - ((b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a*c*x^3) + ((5*b^2*c^2
- 2*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(96*a^2*c^2*x^2) - ((b*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d
+ 15*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(192*a^3*c^3*x) + ((b*c - a*d)^2*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^
2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/2)*c^(7/2))

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Rubi [A]  time = 0.194992, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {97, 151, 12, 93, 208} \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right )}{96 a^2 c^2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{192 a^3 c^3 x}+\frac{\left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{24 a c x^3}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^5,x]

[Out]

-(Sqrt[a + b*x]*Sqrt[c + d*x])/(4*x^4) - ((b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(24*a*c*x^3) + ((5*b^2*c^2
- 2*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(96*a^2*c^2*x^2) - ((b*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d
+ 15*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(192*a^3*c^3*x) + ((b*c - a*d)^2*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^
2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/2)*c^(7/2))

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x} \sqrt{c+d x}}{x^5} \, dx &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 x^4}+\frac{1}{4} \int \frac{\frac{1}{2} (b c+a d)+b d x}{x^4 \sqrt{a+b x} \sqrt{c+d x}} \, dx\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 x^4}-\frac{(b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a c x^3}-\frac{\int \frac{\frac{1}{4} \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right )+b d (b c+a d) x}{x^3 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{12 a c}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 x^4}-\frac{(b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a c x^3}+\frac{\left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^2 c^2 x^2}+\frac{\int \frac{\frac{1}{8} (b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right )+\frac{1}{4} b d \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) x}{x^2 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{24 a^2 c^2}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 x^4}-\frac{(b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a c x^3}+\frac{\left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^2 c^2 x^2}-\frac{(b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^3 c^3 x}-\frac{\int \frac{3 (b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right )}{16 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{24 a^3 c^3}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 x^4}-\frac{(b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a c x^3}+\frac{\left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^2 c^2 x^2}-\frac{(b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^3 c^3 x}-\frac{\left ((b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 a^3 c^3}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 x^4}-\frac{(b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a c x^3}+\frac{\left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^2 c^2 x^2}-\frac{(b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^3 c^3 x}-\frac{\left ((b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{64 a^3 c^3}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{4 x^4}-\frac{(b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{24 a c x^3}+\frac{\left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a^2 c^2 x^2}-\frac{(b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^3 c^3 x}+\frac{(b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.251971, size = 194, normalized size = 0.76 \[ \frac{\frac{3 x^2 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \left (x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+a d x+b c x)\right )}{a^{5/2} c^{5/2}}+\frac{40 x (a+b x)^{3/2} (c+d x)^{3/2} (a d+b c)}{a c}-48 (a+b x)^{3/2} (c+d x)^{3/2}}{192 a c x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^5,x]

[Out]

(-48*(a + b*x)^(3/2)*(c + d*x)^(3/2) + (40*(b*c + a*d)*x*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(a*c) + (3*(5*b^2*c^
2 + 6*a*b*c*d + 5*a^2*d^2)*x^2*(-(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c + b*c*x + a*d*x)) + (b*c
- a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))/(a^(5/2)*c^(5/2)))/(192*a*c*x^4)

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Maple [B]  time = 0.016, size = 705, normalized size = 2.8 \begin{align*}{\frac{1}{384\,{a}^{3}{c}^{3}{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}-12\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-12\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}{d}^{3}+14\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}bc{d}^{2}+14\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{2}{c}^{2}d-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{b}^{3}{c}^{3}+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}c{d}^{2}-8\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}b{c}^{2}d+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}a{b}^{2}{c}^{3}-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}{c}^{2}d-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}b{c}^{3}-96\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}{c}^{3}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^5,x)

[Out]

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*
a*c)/x)*x^4*a^4*d^4-12*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^4*a^3*b*c*d^3
-6*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^4*a^2*b^2*c^2*d^2-12*ln((a*d*x+b*
c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^4*a*b^3*c^3*d+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(
b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^4*b^4*c^4-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*d^
3+14*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b*c*d^2+14*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)*x^3*a*b^2*c^2*d-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*b^3*c^3+20*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*x^2*a^3*c*d^2-8*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^2*b*c^2*d+20*(a*c)^(1/2)*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a*b^2*c^3-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*c^2*d-16*(a*c)^(
1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b*c^3-96*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*c^3*(a*c)^(1/2))/(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)/x^4/(a*c)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 33.1206, size = 1237, normalized size = 4.83 \begin{align*} \left [\frac{3 \,{\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )} \sqrt{a c} x^{4} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \,{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \,{\left (48 \, a^{4} c^{4} +{\left (15 \, a b^{3} c^{4} - 7 \, a^{2} b^{2} c^{3} d - 7 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} c d^{3}\right )} x^{3} - 2 \,{\left (5 \, a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + 5 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \,{\left (a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, a^{4} c^{4} x^{4}}, -\frac{3 \,{\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )} \sqrt{-a c} x^{4} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (a b c d x^{2} + a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \,{\left (48 \, a^{4} c^{4} +{\left (15 \, a b^{3} c^{4} - 7 \, a^{2} b^{2} c^{3} d - 7 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} c d^{3}\right )} x^{3} - 2 \,{\left (5 \, a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + 5 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \,{\left (a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, a^{4} c^{4} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^5,x, algorithm="fricas")

[Out]

[1/768*(3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4*d^4)*sqrt(a*c)*x^4*log((8*a^2
*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) +
8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(48*a^4*c^4 + (15*a*b^3*c^4 - 7*a^2*b^2*c^3*d - 7*a^3*b*c^2*d^2 + 15*a^4*c*
d^3)*x^3 - 2*(5*a^2*b^2*c^4 - 2*a^3*b*c^3*d + 5*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(b*x + a)*
sqrt(d*x + c))/(a^4*c^4*x^4), -1/384*(3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4
*d^4)*sqrt(-a*c)*x^4*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 +
a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) + 2*(48*a^4*c^4 + (15*a*b^3*c^4 - 7*a^2*b^2*c^3*d - 7*a^3*b*c^2*d^2 + 15*a^4
*c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 - 2*a^3*b*c^3*d + 5*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(b*x +
a)*sqrt(d*x + c))/(a^4*c^4*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b x} \sqrt{c + d x}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(1/2)*(d*x+c)**(1/2)/x**5,x)

[Out]

Integral(sqrt(a + b*x)*sqrt(c + d*x)/x**5, x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^5,x, algorithm="giac")

[Out]

Exception raised: TypeError